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\newtheorem{theorem}{Theorem}
\newtheorem{conjecture}{Conjecture}
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\title{Geometric rigidity of graphs on the torus}

\author{Elissa Ross, Brigitte Servatius and Herman Servatius }
\address{Mathematical Sciences, Worcester Polytechnic Institute, Worcester MA 01609-2280}
\email{bservat@wpi.edu}

\date{\today}


\begin{document}
\maketitle


%\section*{Geometric rigidity on the fixed torus}

A {\it gain graph} is a graph $G$ whose edges are labeled invertibly by the elements of a group $\Gamma$ \cite{Zaslavsky}. We denote a gain graph by $\langle G, g \rangle$, where $g: E(G)^+ \rightarrow \Gamma$ is the labelling of the forward-directed edges.
If the directed edge $e$ has label $g_e$, then the other direction of the edge has label $g_e^{-1}$ (see Figure \ref{fig:gain}).
When $\Gamma = \mathbb Z^2$, the gain graph $\langle G, g \rangle$, together with a {\it realization} $p$ of the vertices $V(G)$ in the plane, provides a description of an infinite periodic graph realized in $\mathbb R^2$ (see Figure 2).
We view $p$ as a realization of the vertices on the {\it fixed torus}: the topological torus formed by identifying the opposite sides of the square $[0,1] \times [0,1]$.


The present work is concerned with three objects:
\begin{enumerate}
	\item An infinite periodic graph, realized in $\mathbb R^2$ ({\it infinite periodic framework}). Denote by $(\langle G, g \rangle, p)$.

	\item Its underlying {\it gain graph:} $\langle G, g \rangle$, where $G$ is the quotient graph under
translational symmetry, and $g$ is a labelling of the edges by elements of $\mathbb Z^2$.

	\item The {\it geometric gain framework}: The quotient graph $G$ realized in $\mathbb R^2$ with same
vertex geometry as in the infinite periodic graph, but now with straight edges, and no gains. Denote by $(G,p)$.
\end{enumerate}

%Notation: Denote the infinite periodic framework as $(\langle G, g \rangle, p)$, and denote the geometric gain framework by $(G,p)$.

An {\it infinitesimal motion} of an infinite periodic framework is an assignment $u: V(G) \rightarrow \mathbb R^2$ of infinitesimal velocities to the vertices of the gain graph $\langle G, g \rangle$ on the fixed torus such that the edges lengths of the framework are (infinitesimally) preserved. If an infinitesimal motion preserves the distances between {\it all} pairs of vertices, it is called {\it trivial}. Such an infinitesimal motion is forced to be {\it periodic}, since it is defined on orbits of vertices and edges in the quotient graph.

\begin{theorem}
Fix one vertex of the infinite periodic framework, and fix the corresponding vertex of
the geometric gain graph. If there is a nontrivial infinitesimal motion of the
infinite periodic framework $(\langle G, g \rangle, p)$, then we can ``shrink" the  gains
to obtain an infinitesimal motion of the geometric gain framework $(G,p)$. The infinitesimal motion
of the geometric gain framework is either a rotation about the fixed vertex, or is non-trivial.
\end{theorem}

In other words, every infinitesimal motion of the infinite periodic framework corresponds to an infinitesimal motion
of the geometric gain framework. The converse is not true. However, it is an interesting question to determine which motions of the geometric gain graph
lift to motions of the infinite periodic framework.

The following example of a generically rigid graph which is geometrically flexible provides an
illustration of our methods. By \cite{Ross} $K_4$ is generically rigid on the fixed torus. In the geometric position with coordinates given below,
it is flexible. We show that the infinitesimal motion on the periodic framework can be shrunken to an
infinitesimal motion of the geometric gain graph.

\begin{figure}
\centering
\includegraphics[scale=.65]{abgrid00a}\qquad
\includegraphics[scale=.65]{abgrid00ax}
\caption{Gain graph $\langle G, g \rangle$. Unlabeled edges have gain $(0,0)$. \label{fig:gain}
Red arrows indicate an infinitesimal rotation lifting to $(\langle G, g \rangle, p)$ }
\end{figure}
\begin{figure}
$$\includegraphics[scale=.45]{abgrid02a} \quad \includegraphics[scale=.45]{abgrid01a}$$
\caption{The periodic framework corresponding to the gain graph of Figure \ref{fig:gain} (left). 
The ``shrink" of this infinitesimal motion (red arrows) in progress (right). }
\end{figure}

Let the vertices be placed as follows:
\[
p_0 = (0.5, 0),
\  p_1 = (0,0),
\ p_2 = (0, 0.5),
\ p_3 = (0.5, 0.5).
\]
Let $g_1 = (1,0)$, $g_2 = (-1,0)$.
The initial motion assignment $u$ is:
\[u = (\begin{array}{cccccccc} 0 & 0 & 0 & -a & a & -a & a & 0 \end{array}).\]
%Or in other words $u_0 = (0,0), \ u_1 = (0, -a), \ u_2 = (a, -a), \ u_3 = (a, 0)$.

We shrink this motion as follows: for $t \in [-1,1]$, let $g_1$ and $g_2$ be written in homogeneous
coordinates as $g_1(t) = [\frac{1-t}{2}: 0 :1]$ and $g_2(t) = [0: \frac{1-t}{2} : t]$. Note that as
$t$ passes through zero, the $y$ coordinate of $g_2(t)$ passes through infinity. If we demand that
$u_0(1) = u_0(-1)$, and $u_3(1) = u_3(-1)$ (i.e. that the initial velocities at the vertices
$v_0$ and $v_3$ remain fixed), then as we shrink the gains, the infinitesimal motions at the
vertices $v_1$ and $v_2$ are as follows:
\[u(t) = (\begin{array}{cccccccc} 0 & 0 & 0 & at & a & at & a & 0 \end{array}).\]
In particular, when $t=1$ and $g_1(1) = g_2(1) = (0,0)$, then
\[u(1) = (\begin{array}{cccccccc} 0 & 0 & 0 & a & a & a & a & 0 \end{array}),\]
which is an infinitesimal rotation of $(\langle G, g \rangle, p)$ about the fixed vertex $p_0$.

\begin{thebibliography}{9}

\bibitem{Ross}
  Elissa Ross,
  \emph{Geometric and combinatorial rigidity of periodic frameworks as graphs
 on the torus}.
Ph.D. Thesis York University (Canada).
ProQuest LLC, Ann Arbor, MI,  2011. 336 pp. ISBN: 978-0494-75689-8

\bibitem{Zaslavsky}
T.~Zaslavsky.
\newblock Biased graphs. I. Bias, balance, and gains.
\newblock {\em J. Combin. Theory Ser. B}, 47:32 -- 52, 1989.




\end{thebibliography}

\end{document}
